# From optimal to investable: implementing a portfolio of ETFs, part 2

--

Note: A Google Sheets spreadsheet corresponding to this story is available here.

In the previous article of this series, the difference between an *optimal* portfolio¹ and an *investable* portfolio² was illustrated with a portfolio of ETFs.

It was also established that this difference is a problem that only plagues small portfolios.

# Diversified investable portfolios

Unfortunately, large portfolios are subject to other problems, and in particular to the problem of concentration risk.

For example, if you are implementing a **60/40 Portfolio** strategy — with 60% of your portfolio invested in US stocks and 40% of your portfolio invested in US bonds — , you might be using the standard IVV and TLT ETFs from the ETF provider iShares.

Problem is, if you were to invest several million dollars in such a portfolio, you would actually be trusting only one ETF provider with all these dollars!

Graphical example below, for a 2,500,000 USD portfolio:

Such a portfolio presents a very high concentration risk in terms of ETF providers.

Hopefully, there is a simple way to alleviate this risk, which is to incorporate ETFs from other ETF providers:

- Alternatives to IVV: SPY (SPDR), VOO (Vanguard)
- Alternatives to TLT: SPTL (SPDR), VGLT (Vanguard)

Graphical example below, for an ETF provider-diversified 2,500,000 USD portfolio:

While as “close” to the optimal **60/40 Portfolio** as its 100% iShares sibling, this investable portfolio bears far less concentration risk in terms of ETF providers thanks to its perfect diversification among 3 different ETF providers.

# Mathematical formulation of the problem

In real life, the problem of minimizing concentration risk in a large portfolio of ETFs is more complex than the toy example above:

- While it might be desirable to diversify ETF providers, an ETF tracking a specific asset class might only available from a single ETF provider
- In addition to diversifying ETF providers, it might also be desirable to diversify synthetic v.s. physical ETFs³, or to diversify distributing v.s. accumulating ETFs⁴, or to diversify other intrinsic ETFs characteristics
- Going beyond intrinsic ETFs characteristics, it might also be desirable to diversify brokerage firms⁵, currencies exposition, etc.

In all generality, the problem of minimizing concentration risk in an investable portfolio can be formalized as a mathematical optimization problem whose aim is to

compute the investable portfolio which exhibits the minimum deviation⁶ from the optimal portfolio while best⁷ taking into account concentration risk constraints.

# Computational solution

As in the previous article of this series, solving this optimization problem is computationally very challenging, so that the usage of heuristics is required to compute a *good enough *solution within a reasonable time frame.

Such a heuristic, using a stochastic optimization algorithm, is provided by **Portfolio Optimizer** through a Web API.

Re-using the example above of a **60/40 Portfolio **of 2,500,000 USD, an investable portfolio satisfying maximum ETF providers concentration constraints can easily be computed by **Portfolio Optimizer**:

- Through cURL:

curl "https://api.portfoliooptimizer.io/v1/portfolio/construction/investable" \

-H "Content-Type: application/json" \

-d '{ "assets": 6,

"assetsGroups": [[1,2,3],[4,5,6],[1,6],[2,4],[3,5]],

"assetsGroupsWeights": [0.6,0.4,null,null,null],

"maximumAssetsGroupsWeights:" [null,null,0.25,0.25,0.50],

"assetsPrices": [378.65,380.05,348.1,152.31,92.69,43.63],

"portfolioValue": 2500000 }'{

"assetsPositions":[1045,1041,2035,1504,5844,5256],

"assetsWeights":[0.1582,0.1584,0.2833,0.0916,0.2166,0.0916]

}

- Through a Google Sheets integration

This concludes this 2-part series on the computation of investable portfolios of ETFs.

[1]: Made of real-valued weights, like 86.42%

[2]: Made of integer-valued number of shares, like 5 shares

[3]: The underlying risks are not the same

[4]: Typically, for taxes reasons

[5]: Typically, to reduce the risk of a single broker going bankrupt, or the risk of having a portfolio fully or partially frozen for whatever reason linked to a single broker (technical maintenance, trading restrictions suddendly imposed like Robinhood did with the GME stock…)

[6]: An example of measure of the deviation between two portfolios is the Euclidean distance between the two portfolios real-valued weights

[7]: The word “best” is important here, because — mathematically — an investable portfolio satisfying all the risk concentration constraints might not exist.

*Disclaimer*

*Under no circumstances does any information on this story represent a recommendation to buy, sell, or hold any security.*

*None of the content of this story is guaranteed to be correct, and anything shown should be subject to independent verification.*

*You, and you alone, are solely responsible for any investment decisions that you make.*